(navigation image)
Home American Libraries | Canadian Libraries | Universal Library | Community Texts | Project Gutenberg | Biodiversity Heritage Library | Children's Library | Additional Collections
Search: Advanced Search
Anonymous User (login or join us)
See other formats

Full text of "Men Of Mathematics"

definite order, corresponding to 'numberable' properties of the
elements, the assignment in the given order corresponding to a
preassigned ordering of the 'numberable* properties. Granted
that this may be even less comprehensible than Riemann's
definition, it is nevertheless a -working basis from which to
start, and all that it amounts to in plain mathematics is this:
a manifold is a set of ordered 'n-tuples' of numbers (aj1Ja?s ... 5
#), where the parentheses, (), indicate that the numbers
lsa22, ..., xn are to be written in the order given. Two such
tt-tuples, (o!lyr2,... , xn) and (y-&& ... , yn) are equal when, and
only when, corresponding numbers in them are respectively
equal, namely, when, and only when, a^ = ylt #2 = yzi ..,}
*n = !/
If precisely n numbers occur in each of these ordered n-tuples
in the manifold, the manifold is said to be of n dimensions. Thus
we are back again talking co-ordinates with Descartes. If each
of the numbers in (ajlsa2, ... , xn) is a positive, zero, or negative
integer, or if it is an element of any countable set (a set whose
elements may be counted off 1,2,3, ...), and if the like holds for
every n-tuple in the set, the manifold is said to be discrete. If
the numbers xvx, ... , xn, may take on values continuously
(as in the motion of a point along a line), the manifold is
This working definition has ignored - deliberately - the ques-
tion of whether the set of ordered n-tuples is 'the manifold' or
whether something "represented by' these is 'the manifold'.
Thus, when we say (x,y) are the co-ordinates of a point in a
plane, we do not ask what 'a point in a plane' is, but proceed to
work with these ordered couples of numbers (x%) where x,y run
through all real numbers independently. On the other hand it
may sometimes be advantageous to fix our attention on what
such a symbol as (x,y) represents. Thus if # is the age in seconds
of a man and y his height in centimetres, we may be interested
in the man (or the class of all men) rather than in his co-ordinates
with which alone the mathematics of our enquiry is concerned. In
this same order of ideas, geometry is no longer concerned with
what 'space' 'is' -whether 'is' means anything or not in relation
to 'space1. Space, for a modern mathematician, is merely a